Practice | Hashtable

11. If log 2 = 0.30103, the number of digits in 2⁶⁴ is:

A. 18

B. 19

C. 20

D. 21

Solution:

\begin{aligned} \log (2^{64}) \\ = 64 \times \log 2 \\ = (64 \times 0.30103) \\ = 19.26592 \end{aligned}

Its characteristic is 19.
Hence, then number of digits in 2⁶⁴ is 20.

12. If $\log_{x} (\frac{9}{16}) = - \frac{1}{2},$ then x is equal to:

A. $- \frac{3}{4}$

B. $\frac{3}{4}$

C. $\frac{81}{256}$

D. $\frac{256}{81}$

Discuss

Solution:

\begin{aligned} \log_{x} (\frac{9}{16}) = - \frac{1}{2} \\ \Rightarrow x^{- \frac{1}{2}} = \frac{9}{16} \\ \Rightarrow \frac{1}{\sqrt{x}} = \frac{9}{16} \\ \Rightarrow \sqrt{x} = \frac{16}{9} \\ \Rightarrow x = {(\frac{16}{9})}^{2} \\ \Rightarrow x = \frac{256}{81} \end{aligned}

13. If a^x = b^y, then:

A. $\log \frac{a}{b} = \frac{x}{y}$

B. $\frac{\log a}{\log b} = \frac{x}{y}$

C. $\frac{\log a}{\log b} = \frac{y}{x}$

D. None of these

Discuss

Solution:

\begin{aligned} a^{x} = b^{y} \\ \Rightarrow \log a^{x} = \log b^{y} \\ \Rightarrow x \log a = y \log b \\ \Rightarrow \frac{\log a}{\log b} = \frac{y}{x} \end{aligned}

14. If log_x y = 100 and log₂ x = 10, then the value of y is:

A. 2¹⁰

B. 2¹⁰⁰

C. 2¹⁰⁰⁰

D. 2¹⁰⁰⁰⁰

Discuss

Solution:

\begin{aligned} \log_{2} x = 10 \Rightarrow x = 2^{10} \\ ∴ \log_{x} y = 100 \\ \Rightarrow y = x^{100} \\ \Rightarrow y = {(2^{10})}^{100} [put value of x] \\ \Rightarrow y = 2^{1000} \end{aligned}

15. The value of log₂ 16 is:

A. $\frac{1}{8}$

B. 4

C. 8

D. 16

Discuss

Solution:

Let log₂ 16 = n.
Then, 2ⁿ = 16 = 2⁴⇒ n = 4
∴ log₂ 16 = 4

16. The value of $\log_{5} \frac{(125) (625)}{25}$ is equal to -

A. 725

B. 5

C. 3125

D. 6

Discuss

Solution:

\begin{aligned} lo g_{5} \frac{(125) (625)}{25} \\ = lo g_{5} (\frac{5^{3} \times 5^{4}}{5^{2}}) \\ = lo g_{5} 5^{5} \\ = 5lo g_{5} 5 \\ = 5 \end{aligned}

17. Determine the value of $lo g_{3 \sqrt{2}} (\frac{1}{18})$ is = ?

A. 2

B. -2

C. $\sqrt{2}$

D. $\sqrt{3}$

Discuss

Solution:

\begin{aligned} lo g_{3 \sqrt{2}} (\frac{1}{18}) \\ = lo g_{3 \sqrt{2}} (\frac{1}{{(3 \sqrt{2})}^{2}}) \\ = lo g_{3 \sqrt{2}} {(3 \sqrt{2})}^{- 2} \\ = (- 2) lo g_{3 \sqrt{2}} 3 \sqrt{2} \\ = - 2 \end{aligned}

18. The value of $lo g_{10} (0.0001)$ is = ?

A. $\frac{1}{4}$

B. $- \frac{1}{4}$

C. $- 4$

D. $4$

Discuss

Solution:

\begin{aligned} lo g_{10} (0.0001) \\ = lo g_{10} (\frac{1}{10000}) \\ = lo g_{10} (\frac{1}{10^{4}}) \\ = lo g_{10} 10^{- 4} \\ = - 4 lo g_{10} 10 \\ = - 4 \end{aligned}

19. What is the value of ${[lo g_{10} (5lo g_{10} 100)]}^{2}$ = ?

A. 1

B. 2

C. 10

D. 25

Discuss

Solution:

\begin{aligned} {[lo g_{10} (5lo g_{10} 100)]}^{2} \\ = {[lo g_{10} {5lo g_{10} {(10)}^{2}}]}^{2} \\ = {[lo g_{10} (5 \times 2)]}^{2} \\ = {(lo g_{10} 10)}^{2} \\ = 1 \end{aligned}

20. If $lo g_{8} p = 25$ and $lo g_{2} q = 5,$ then -

A. p = q¹⁵

B. p² = q³

C. p = q⁵

D. p³ = q

Discuss

Solution:

\begin{aligned} lo g_{8} p = 25 and lo g_{2} q = 5 \\ \Rightarrow p  = 8^{25} and q = 2^{5} \\ \Rightarrow p  = {(2^{3})}^{25} and q  = 2^{5} \\ \Rightarrow p  = 2^{75} and q  = 2^{5} \\ \Rightarrow p  = {(2^{5})}^{15} and q  = 2^{5} \\ \Rightarrow p  = q^{15} \end{aligned}